Problem: Simplify the following expression and state the condition under which the simplification is valid. $z = \dfrac{-5a^2 - 35a - 30}{8a^3 - 40a^2 - 48a}$
First factor out the greatest common factors in the numerator and in the denominator. $ z = \dfrac {-5(a^2 + 7a + 6)} {8a(a^2 - 5a - 6)} $ $ z = -\dfrac{5}{8a} \cdot \dfrac{a^2 + 7a + 6}{a^2 - 5a - 6} $ Next factor the numerator and denominator. $ z = - \dfrac{5}{8a} \cdot \dfrac{(a + 1)(a + 6)}{(a + 1)(a - 6)}$ Assuming $a \neq -1$ , we can cancel the $a + 1$ $ z = - \dfrac{5}{8a} \cdot \dfrac{a + 6}{a - 6}$ Therefore: $ z = \dfrac{ -5(a + 6)}{ 8a(a - 6)}$, $a \neq -1$